Optimal new product design using reverse quality function deployment with nonlinear regression modeling

Abstract

Purpose

Quality function deployment (QFD) has been widely applied in new product development, but existing research on QFD has some limitations. Primarily, QFD lacks the capability to provide feedback on the satisfaction degree of customer requirements (CRs) according to the actual values of engineering characteristics (ECs). In addition, QFD does not quantitatively consider the interrelationships among ECs. Reverse QFD (R-QFD) was introduced to implement the feedback process. On this basis, this paper quantitatively considers the interrelationships among ECs in the R-QFD model and extends these relationships to encompass combinations of multiple ECs, aiming to improve the inference accuracy of the model.

Design/methodology/approach

A nonlinear regression model was established between CRs and ECs, aiming to infer the satisfaction degree of CRs based on the implementation status of ECs. This model considers the interdependencies among ECs and extends the consideration of pairwise EC correlations from every two to every fifteen. Lingo Software is utilized to seek solutions for this program. To facilitate the implementation of the program, a directive to simplify the solution has been proposed.

Findings

The experimental results indicate that the interrelationships among ECs significantly affect the inference accuracy of the R-QFD model, thereby verifying the necessity of considering higher-order interrelationships among ECs within the R-QFD framework. Based on the results from data experiments, this paper also proposes research recommendations pertaining to ECs hierarchy for varying quantities of ECs.

Originality/value

The outcomes of this study have further refined the R-QFD model, addressing its limitations of ignoring the interrelationships among ECs. This transformation elevates the R-QFD model from a relatively simple linear model to a nonlinear model formed through modeling, thereby enhancing its accuracy and applicability. In practical terms, this study provides case support for the application of the R-QFD model in manufacturing industry.

Full text

Translate

Turn on search term navigation

1. Introduction

Quality function deployment (QFD) was proposed by Japanese quality management scholars Yoji Akao and Shigeru Mizuno in the early 1970s, and has been widely applied in the world wide. QFD can transform customer requirements (CRs) to engineering characteristics (ECs) in product design and development and further to the remainder processes of the production and delivery processes, i.e. production process development, manufacturing, logistics (Akao, 2012). The tool of QFD looks like a house and is also called as house of quality (HOQ). The CRs and their weights are located in the left wall; the ECs and the interrelationships between each pair of ECs are located in the top roof; the relationship matrix between CRs and ECs is located in the center; the competitive evaluation of CRs is located in the right wall; the competitive evaluation of ECs is located in the basement. There are six basic steps for building the HOQ, i.e. step (1) identify CRs and determine their weights, step (2) identify ECs, step (3) relate CRs to ECs, step (4) conduct an evaluation of competing products or services, step (5) evaluate ECs and develop targets, step (6) determine which ECs to deploy in the remainder of the production and delivery processes (Evans and Lindsay, 2011). In step 5, the priorities of CRs can be transformed to the weights of ECs through a multiplication operation with the relationship matrix between CRs and ECs. The priorities of ECs are the targets of ECs to be deployed in the remainder processes, and can be decided according to the weights of ECs, the competitive evaluations and the interrelationships analysis among ECs. The sequence of the above 6 steps was understood as the forward process in QFD.

The priorities of ECs decided by QFD will be deployed in the remained process. However, due to the limited resources in the remainder process, some priority ECs may not be completely supported. In these cases, how the targeted CRs would be realized in the new design needs to be examined. Usually, market research can be used to check the CRs satisfaction degree of newly designed products. However, this approach tends to incur substantial time costs, thereby hindering the expeditious pace of new product development for enterprises. Meanwhile, this approach makes it easier to illustrate the innovations of new design to competitors, thereby precipitating the risk of design emulation. A feedback process was defined as speculating the satisfaction degree of CRs according to the actual values of ECs by the following processes in practice. This process is called the feedback process of product design in this paper. The forward process and the feedback process can be shown as Figure 1. Wang et al. (2020) provided a clear elucidation of the feedback process and established the R-QFD method. R-QFD can quickly obtain the satisfaction degree of CRs based on the actual values of ECs without market survey on CRs. With R-QFD, QFD processes can be changed from an open-loop process to a closed-loop process so as to improve customer satisfaction (CS).

The research on the practical application of QFD reveals two notable deficiencies. Firstly, when ECs cannot be fully realized in the remainder processes, QFD fails to provide feedback on the satisfaction degree of CRs. Wang et al. (2020) addressed this gap by introducing the R-QFD model. However, the limited studies on R-QFD have primarily focused on the feedback process from ECs to CRs, neglecting other influential factors within QFD, i.e. competitive evaluation. Secondly, the interrelationships among ECs have a significant impact on both QFD and R-QFD, but this has not been given sufficient attention, which affects the accuracy of the conversion process. Dawson and Askin (1999) highlighted that when there are a large number of ECs that are difficult to be deployed quantitatively, there is a lack of research on the impact of interrelationships. On this basis, this study extended the R-QFD by establishing a mathematical model to analysis the impact of the interrelationships among ECs. This paper systematically describes the general procedural steps of the R-QFD process, thereby providing a clearer and more referable framework for subsequent R-QFD research work. The optimized R-QFD model, which incorporates the interrelationships among ECs, enhances its proximity to the actual realization status of objectives. Consequently, this model provides decision-makers with more effective managerial tools.

The following sections were organized as follows. Literature review was summarized in Section 2. In Section 3, the 5 steps in the operation process of R-QFD were indicated in detail. And the operation process flowchart of QFD and R-QFD was proposed in the basis of the quantitative interrelationships among ECs. In Section 4, the optimization model in R-QFD was established, and solution of the optimization model was discussed. The numerical experiment was designed to prove the feasibility of the optimization model. In Section 5, an instruction was proposed to simplify the solution process through the numerical experiments. In Section 6, the conclusions and future work were summarized.

2. Literature review

2.1 Researches on CRs capture and priorities

CRs come from the various customers with different background. It is difficult to unify their expressions. In order to deal with the inaccurate expression on CRs, fuzzy estimation methods have often been adopted in QFD. Khoo and Ho (1996) presented an improved method called as fuzzy quality function deployment (FQFD). The FQFD applied the possibility theory and fuzzy arithmetic to address the ambiguity in estimation. Temponi et al. (1999) developed an extended HOQ with fuzzy logic to capture the imprecise CRs. A formal representation of requirements and heuristic reasoning scheme were adopted to infer the implicit relationships among CRs. Bottani and Rizzi (2006) adopted fuzzy logic to deal with the ill-defined nature of the qualitative linguistic judgments in QFD so as to effectively and efficiently improve logistics performance and CS. Kwong and Bai (2002) proposed to use fuzzy numbers to determine the weights of CRs in QFD. Although the applications of fuzzy methods in QFD can solute the problem of language inaccuracy, it also reduces the accuracy of CRs’ capture and priorities. Otherwise, design of experiment (DOE) is also one of the commonly used methods of quality engineering research. Li et al. (2013) considered the limitations of CRs capture in QFD and proposed that conjoint analysis and discrete choice experiments (DCE, an extension of DOE) can contribute immensely to quality improvement by enhancing the evaluation of the voice of customer (VOC), but their case only considered a small amount of ECs. It is disadvantage to design a new product with large number of ECs.

2.2 Researches on ECs priority estimation

Fuzzy estimation methods have often been applied on generating priority of ECs in QFD. Wang (1999) introduced multi-criteria decision methods in QFD and proposed a new fuzzy outranking approach to prioritize ECs. Kahraman et al. (2006) proposed an integrated framework based on fuzzy-QFD and a fuzzy optimization model to determine ECs requirements. Wu and Liao (2018) introduced a multi-expert multi-criteria decision method in QFD with a complicated fuzzy linguistic representation model, i.e. probabilistic linguistic term set (PLTS), and a ranking method, i.e. ORESTE. Karsak et al. (2003) presented a 0–1 goal programming model to determine the weights of ECs in QFD with analytic network process (ANP), but the model did not consider the interrelationships among ECs.

2.3 Researches on relationship matrix between CRs and ECs

Fuzzy methods are also the mainstream method to determine the planning matrix. Vanegas and Labib (2001) proposed a novel method for determining optimum targets in QFD, which use fuzzy numbers to represent the imprecise nature of the judgments, and to define more appropriately the relationships between ECs and CRs. Considering that the product design process is carried out in an uncertain environment, Chen and Weng (2006) adopted a fuzzy method to formulate the relationships between CRs and ECs. And they proposed a fuzzy target planning model to determine requirements. In addition, some scholars incorporating Kano’s model into the planning matrix of QFD to help accurately and deeply understand the nature of the VOC (Tan and Shen, 2000). The Kano’s model is a useful tool for classifying and prioritizing CRs. According to the relationship between different types of quality characteristics and CR, Noriaki Kano divides the quality characteristics of products and services into five categories: must-be quality; one-dimensional quality; attractive quality; indifferent quality; reverse quality. However, it is only analyzed from a qualitative point of view, and it had not been well quantified.

2.4 Researches on QFD optimization models

Kim et al. (2000) presented an integrated formulation and solution method on QFD. The method built the fuzzy regression and fuzzy optimization models and defined the model components, i.e. parameters, objectives, and constraints, in a crisp or fuzzy way based on multi-attribute value theory. Chen and Chen (2006) built a non-linear programming model on QFD based on the possibility regression models. Tang et al. (2002) presented a fuzzy formulation solution combined with a genetic-based interactive approach in QFD. This solution built the fuzzy optimization models based on the components of QFD, i.e. the planned degree and actual achieved degree in CRs and ECs, the required primary costs and actual costs. The above studies used fuzzy estimation methods in QFD, however, the interrelationships among ECs have not been examined quantified.

2.5 Researches on integrating QFD with other methods

Yamashina et al. (2002) proposed a new method, named innovative product development process (IPDP), to integrate QFD with Teoriya Resheniya Izobreatatelskikh Zadatch (TRIZ). Moskowitz and Kim (1997) developed a novice-friendly decision support system prototype for QFD-QFD optimizer. It can generate the feasible design solutions quickly, and benefit to understand the complex relationship matrix between each CR and each EC, and the interrelationships among ECs. Bhattacharya et al. (2005) delineated an integrated model combining analytic hierarchy process (AHP) and QFD for the industrial robot selection problems. Many studies can be found on combining fuzzy AHP with QFD and other methods (Kubler et al., 2016). However, few studies can be found on the combination between fuzzy estimation methods and R-QFD.

2.6 Researches on R-QFD

In the research field of the QFD feedback process, Wang et al. (2020) systematically elucidated and consolidated the reverse process, presenting an innovative R-QFD method tailored explicitly for the intricacies of the feedback process. Meanwhile, this study introduced an optimization model in the steps of the R-QFD method, aiming to infer the satisfaction degree of CRs from the actual values of ECs. Applying this approach, Wang et al. (2020) quantified the implementation status of China’s manufacturing 2025 initiative and compared it with the targets. Wang et al. (2020)’s research provided a quantitative assessment of the Manufacturing 2025 initiative, addressing the theoretical gap in the QFD feedback process. However, this model has certain limitations, as the actual satisfaction degree of CRs is computed through multiplying ECs by the generalized inverse matrix of the EC-CR relationship matrix, without considering the interdependencies among ECs. Additionally, QFD is mainly used in product design and development processes and has a wide range of applications in the manufacturing and software development industries, but Wang et al. (2020)’s method lacks supportive case studies in these areas.

The related studies on QFD and R-QFD are summarized as shown in Table 1. In summary, after the creation of the QFD method, fuzzy estimation methods have been widely employed in various processes, including the identification and ranking estimation of CRs, ranking estimation of ECs, establishment of the relationship matrix between CRs and ECs, and optimization modeling. Fuzzy methods facilitate the transformation of vague and uncertain information into measurable input data, addressing the issue of linguistic inaccuracies within the QFD model. However, the transformation process may not capture the entirety of information, leading to potential inaccuracies in the results. Alternative approaches, such as the Kano model, primarily focus on qualitative aspects and lack quantitative analysis. Furthermore, existing models predominantly emphasize the forward process of QFD, neglecting the reverse research perspective that starts from the actual values of ECs to estimate the satisfaction degree of CRs. Wang et al. (2020) proposed the R-QFD model but did not consider that the interrelationships among ECs would have a huge impact on the feedback process. Building upon the R-QFD model established by Wang et al. (2020), this study will focus on the quantification of the interrelationships among ECs in the R-QFD process. R-QFD is helpful to improve different CRs simultaneously, especially facing the trade-off interrelationships among ECs. The case drawn from the Japanese manufacturing industry employed in this study can offer practical validation for the R-QFD model. Additionally, the refined R-QFD model proposed in this paper facilitates sensitivity analysis, supporting decision-making for ECs under constraints such as time or investments.

3. Operation process of QFD and R-QFD

The QFD process is a forward process, while the R-QFD process is a feedback process. QFD and R-QFD form a closed-loop process, where certain steps and data in their respective processes are common or comparable. Therefore, this section first briefly introduces the QFD process, and then elaborates on the R-QFD process in detail on this basis.

The parameters in traditional HOQ as shown in Figure 2 are explained as follows (Chen and Chen, 2006).

CRs, referred to $C R = {({C R}_{i})}_{n \times 1}$ ;
Weights of CRs, referred to $W C = {({w c}_{i})}_{n \times 1}$ ;
Competitive evaluation: comparison analysis between own CRs or ECs and the competitors’ CRs or ECs.
ECs, referred to $E C = {({E C}_{j})}_{m \times 1}$ ;
Weights of ECs, referred to $W E$ = ${({w e}_{j})}_{m \times 1}$ ;
Relationships matrix between CRs and ECs: to what extend the ECs impact on CRs, referred to $R = {(r_{i j}^{'})}_{n \times m}$ ;
The interrelationships among ECs, referred to $T = {(t_{k j})}_{m \times m}$ .

Traditionally, there are six basic steps in QFD operations including identify CRs, identify ECs, etc. The operation process flowchart of QFD can be summarized as the following Figure 3. The introduction on each process can refer to the textbook (Evans and Lindsay, 2011).

Based on the same parameter setting, the operational process flowchart of R-QFD can be summarized as Figure 3. There are 5 steps in the operation process flowchart of R-QFD: (1) implement the plan; (2) collect the actual values of ECs; (3) get the satisfaction degree of CRs; (4) compare with the target values of CRs; (5) revise or output the design plan.

1) Implement the plan

In the product design process, the first plan is generally the output result of QFD, and the subsequent plan is a revised plan based on the output result of R-QFD. The design plan here is no longer limited to the forward process in product design, but also considers the feedback process in product design.

2) Collect the actual values of ECs (the input data of R-QFD)

Five types of data were collected as the input data of R-QFD, including: (1) the target values of CRs; (2) the planned ECs that are design characteristics that describe the CRs as expressed in the language of the designer or engineer. Essentially, they are the “HOWs” by which the company will respond to the “WHATs” – CRs. They must be measurable, because the output is controlled and compared to objective targets; (3) the correlation matrix between CRs and ECs; (4) the interrelationship matrix between each pair of ECs.

The interrelationships among ECs have a dramatic impact on the satisfaction degree of CRs. If not addressed, dependence results in a duplication of effort and sub-optimal ECs values. The interrelationships matrix among ECs is typically coded in accordance with the subjective measures ${□, Δ, ○, ●, ▲, ■} = {- 9, - 3, - 1, 1, 3, 9}$

Normally, the interrelationship matrix scoring needs to be normalized. The value of normalized matrix is between 1 and −1. It is obvious from statistics that the normalized covariance matrix is the correlation matrix of the original data. This step is absolutely significant. Normalization allows data to eliminate dimensions and make it comparable. If normalization is not carried out, it will often have a relatively large impact on the results of the model solution. Therefore, the following research in the paper will be based on the normalized covariance matrix of the score matrix among ECs to show their interrelationships.

The above four types of data are the same as the data used in generic QFD process. For R-QFD, the fifth type of data is the actual values of ECs. It is derived from the research of Wang et al. (2020), which means the actual implementation of the design plans. Because not all design requirements can be completely satisfied in the following process after design, it is necessary to estimate how the planned ECs are satisfied. The actual value reached by the current ECs can be obtained to provide feedback on the following process in order to correct the deviation as soon as possible.

3) Get the satisfaction degree of CRs (the output data of R-QFD)

Based on the information collected in the former two steps, it is possible to speculate the satisfaction degree of CRs based on the actual values of ECs. In third step of QFD process called “relate CRs to ECs,” CRs is usually multiplied by the relationship matrix between CRs and ECs to describe ECs. However, this method fails to encapsulate the interrelationships among ECs. Similarly, in the R-QFD process, if the inverse operation is directly performed, that is, the actual satisfaction degree of CRs is obtained by multiplying the actual values of ECs with the generalized inverse matrix of the relationship matrix between CRs and ECs, it is not sufficient to reflect the relationships among ECs. Recognizing the importance of considering the interrelationships among ECs, it is necessary to multiply ECs by both the generalized inverse matrix of the relationship matrix between CRs and ECs and the interrelationship matrix among ECs to comprehensively describe CRs. Therefore, an optimization model is proposed for this step. The model will be founded in the next section.

4) Compare with the target value of CRs

This step identifies the actual achievement for each CR and evaluates competitors’ existing products or services for each of them. Customer importance ratings represent the areas of greatest interest and highest expectations as expressed by the customer. Competitive evaluation highlights the absolute strengths and weakness in competing products. With the actual satisfaction degree of CRs, it is possible to compare the actual achieved CRs with the target CRs. Therefore, it is possible to identify which CRs in QFD have achieved their goals through the operation of R-QFD. In addition, with the results of R-QFD, it is helpful to evaluate the competitive advantage comparing with competitors.

5) Revise or output the design plan

According to the judgment of the previous step, if the actual satisfaction degree of CRs can reach the target value, the results of QFD can be tested to be feasible. Otherwise, the QFD process needs to be readjusted until the target value can be reached.

For the readjustment of the design plan, this step is usually accomplished with the output data of R-QFD. It indicates that there exists some faulty measure in QFD. The reason is likely to be that the limited resources cause the ECs to not be completely satisfied. Therefore, it is necessary to readjust the current design plan in combination with the limited resources, the priority of the ECs, and competitive intelligence gathering.

By using this step, designers can discover the improvement results of the design plan in QFD or R-QFD. It also links R-QFD to a company’s strategic vision and indicates priorities for the design process.

4. An optimization model with second-order interrelationships among ECs and solutions

4.1 An optimization model with second-order interrelationships among ECs

An optimization model used in the 3rd step of R-QFD is introduced as follows. This model only considers the interrelationships between every 2 ECs that is called as the second-order interrelationships.(1) $\min F (x) = \sum_{j = 1}^{m} {(b_{j} - a_{j})}^{2}$ $s . t . {\begin{cases} x_{i} = \sum_{j = 1}^{m} r_{j i} a_{j} + \sum_{k < j} \sum_{j = 2}^{m} r_{j i} t_{k j} a_{k} a_{j} & i = 1, 2, \dots, n & (1 - 1) \\ L_{1 i} \leq x_{i} \leq U_{1 i} & i = 1, 2, \dots, n & (1 - 2) \\ L_{2 j} \leq a_{j} \leq U_{2 j} & j = 1, 2, \dots, m & (1 - 3) \\ \sum_{j = 1}^{m} {(b_{j} - a_{j})}^{2} \leq M \sum_{j = 1}^{m} b_{j}^{2} & (1 - 4) \end{cases}$

The notations of the model are explained as follows:

$a_{j}$ = the target value of the ${E C}_{j}$ , $j = 1, 2, \dots, m$ ;
$b_{j}$ = the actual value of the ${E C}_{j}$ , $j = 1, 2, \dots, m$ ;
$x_{i}$ = the satisfaction degree of the ${C R}_{i}$ realized by the actual value of the ${E C}_{j}$ , $i = 1, 2, \dots, n$ ;
$R^{- 1} = {(r_{j i})}_{m \times n}$ = the generalized inverse matrix of relationships matrix between ${E C}_{j}$ and ${C R}_{i}$ ;
$T = {(t_{k j})}_{m \times m}$ = the interrelationship matrix of each ${E C}_{k}$ with respect to each ${E C}_{j}$ ;
$L_{1 i}, U_{1 i}$ = the lower and upper limits of $x_{i}$ ;
$L_{2 j}, U_{2 j}$ = the lower and upper limits of $a_{j}$ ;
$M$ = the limit of goodness of fit.

The decision variables of the optimization model are $x_{i} (i = 1, 2, \dots, n)$ . The optimization objective of the model is to minimize the difference between the actual values and the target values of ECs. The target values of ECs are generated based on the expected values of CRs and the QFD process as shown in Figure 3.

In constraint (1–1), the regression model was founded on the relationships between the target values of ECs and the satisfaction degree of CRs. The regression model considered the interrelationships between each pair of the ECs and was called as the second-order regression model.

Constraint (1–2) represents that the satisfaction degree of ${C R}_{i}$ will not exceed the upper limit, nor will it be lower than the lower limit of the target value of ${C R}_{i}$ .

Constraint (1–3) represents that the target value of ${E C}_{j}$ will not exceed the upper limit, nor will it be lower than the lower limit of the actual value of ${E C}_{j}$ . The values of the coefficients U and L in the model are estimated according to the actual situations of enterprises.

Constraint (1–4) shows that the overall fitting will not exceed M times its optimal fitting. The number of the CRs (n) is often not equal to the number of the ECs (m). If n < m, the model is an over determined equation set, and no contradictory equation solution may be found. The least square method was adopted to search for the solutions.

There are two notations on the matrixes in the optimization model.

1) The interrelationship matrix among ECs

For the interrelationship matrix among ECs ( $T = {(t_{k j})}_{m \times m}$ ), the covariance coefficients ( $t_{k j}$ ) are considered to quantitatively set the amount of change in ${E C}_{k}$ as a result of a unit increase (or decrease) in ${E C}_{j}$ . The diagonal line of the matrix is set as zero, since the autocorrelation is not considered in QFD or R-QFD. Such interrelationships are initially assumed to be strictly linear in traditional QFD process. For example, if there is a high positive correlation between ${E C}_{1}$ and ${E C}_{2}$ , the correlation coefficient is 0.9, which is $t_{12} = 0.9$ . When ${E C}_{2}$ increases by one unit, the change brought to ${E C}_{1}$ is 0.9. This study not only considers the second-order interrelationship between ECs, but also considers higher-order interrelationships such as third-order and fourth-order.

2) The relationship matrix of between ${E C}_{j}$ and ${C R}_{i}$ and its generalized inverse matrix

The relationship matrix between CRs and ECs ( $R = {(r_{i j}^{'})}_{n \times m}$ ) was adopted in QFD. In QFD, the relationships between CRs and ECs are usually represented by the correlation coefficients of CRs with respect to ECs. In R-QFD model established by Wang et al. (2020), the generalized inverse matrix of the relationships between CRs and ECs ( $R^{- 1} = {(r_{j i})}_{m \times n}$ ) was adopted. Based on the generalized inverse matrix and the actual value of ECs, it is possible to infer the satisfaction degree of CRs. This operation essentially constitutes the inverse computation of the QFD process. Thus, the model established by Wang et al. (2020) is inherently linear. This study considers higher-order relationships among ECs, so that the satisfaction degree of CRs can be determined based on the generalized inverse matrix, multiple relationship matrices among ECs, and the actual values of ECs. Therefore, the model established in this study is nonlinear.

Because all ECs will be working together, if there are interrelationships between 2 ECs, there will exist interrelationships among every 3 ECs, every 4 ECs, every 5 ECs, etc. The higher-order regression models are founded for the higher-order interrelationships on the basis of the second-order regression model. For instance, the third-order regression model is added with three-order cross terms on the basis of the second-order regression model, and the rest remain unchanged. The third-order regression model is recorded as Model 2, and higher-order regression models can be deduced by analogy (Model 2, 3, $\dots$ , 14). For example, an example on Model 14 was shown as follows. $\min F (x) = \sum_{j = 1}^{m} {(b_{j} - a_{j})}^{2}$

4.2 Numerical experiments of optimization models in R-QFD

The preceding sections offer a theoretical methodology for checking the feedback process with R-QFD. The purpose of this part is an application of the methodology with problem instances and an evaluation of its performance. Specifically, the issues of relationships matrix (center and roof part) size and density will be examined relative to the HOQ. The study used LINGO software to solve the optimization models. Through specific instances, the study further confirmed the feasibility of the mathematical models. Then the models with different matrix sizes and densities were simulated for many times to analyze the stability of the solution sets.

Background of this instance comes from a real case of a Japanese automobile company. The detailed representations of CRs and ECs were summarized in Table 2. There are 7 CRs and 9 ECs. In detail, the actual values of ECs were the feedback values from implementing the plans designed in QFD, as shown in Table 3. The relationships of ECs with respect to CRs (one 7 × 9 relationship matrix) were illustrated in Table 4, and the generalized inverse matrix were shown in Table 5 respectively. The interrelationship matrix between each pair of ECs (one 9 × 9 interrelationship matrix) were shown in Table 6. Both the scores in Tables 4 and 6 are evaluated by the company’s internal engineers and experts based on the actual operation process.

LINIGO software was adopted for the solution. The solutions of the second-order regression were set as follows in Table 7.

Judging from the solution of the second-order regression model, the mathematical model is feasible. Based on the instance data, the higher-order regression models were founded and solved. LINGO was employed to code and solve the problem instance. In order to facilitate the comparison with the target value of CRs, the satisfaction degree of CRs were normalized to eliminate the dimension. The results of the models with the orders from 2 to 9 in each regression order are summarized in Table 8. For Table 8, N represents the serial number of the optimization model, X represents the satisfaction degree of CRs, and D represents the standard deviations between the solution sets of the adjacent regression model. $D = \sqrt[n]{\frac{\sum_{i = 1}^{n} {(x_{i} - y_{i})}^{2}}{n}}$ , and the deviations in this paper uniformly use the measurement standard. It reported the satisfaction degree of CRs when considering the different effects among the 9 ECs from second-order to ninth-order. The solution sets test the models’ computational feasibility and validity. For the interrelationship problem, the actual values of ECs reflect the impact of dependent ECs with LINGO software.

Figures 4 and 5 show the satisfaction degree and of 7 CRs of the optimization models and the standard deviations D in the above cases respectively. From Table 8 and Figures 4 and 5, it can be seen that the solution tends to be no changes between the sum interrelationships from second-order to third-order and the sum interrelationships from second-order to ninth-order. This indicates that when there exist interrelationships among a certain quantity of ECs, it may hardly change the results of the optimization models. Next section will analysis the results from different larger numbers of ECs so as to find a simple solution for the optimization models with higher-order interrelationships.

In previous studies, for models that consider the interrelationships among ECs, the number of ECs can only be considered at most 3 (Dawson and Askin, 1999). The main reason is that the model adopted DOE method. The excessive number of ECs makes the experiment too complicated to obtain the final solution. In order to simply the complicated process, the studies adopted DOE method only screen the main significant factors. However, if the product design only considers within 3 ECs, it will cause defects in the product design and fail to achieve the target degree of CS degree. Therefore, the number of ECs to be considered in the product design process is actually far more than three. At the same time, the results of the numerical experiment reveal that there is more than 1% deviation between Model 1 and Model 2 when the number of ECs is 9. This indicates that third-order regression model or even higher-order regression model should be considered. Based on the aforementioned two reasons, for the product design, all ECs which affect CS degree should be considered from the perspective of full factor and higher-order regression model should also be consider in R-QFD. This is exactly what the current research institute lacks.

5. A simplified solution for optimization models with higher-order interrelationships

With the establishment of higher-order regression models, the solution process has become more and more complicated. How to simplify the solution process so that the optimization models can be better applied is also a problem to be considered in this paper. When the deviation of the solution set of the adjacent regression order model does not exceed 0.01, the regression order is regarded as the optimal regression order. An instruction will be verified by the numerical experiment. As mentioned above, as the regression order increases, the deviation between the solution sets of the adjacent order regression R-QFD model gradually decreases. In order to give a reference for the best regression order in the case of a specific number of ECs, this part will simulate and summarize the case when the number of ECs is from 10 to 15. Refer to the established mathematical model (Model 9–14), LINGO was used to simulate the case of ECs under different numbers (from 10 to 15).

The simulation of the number of cutoff points 10 ECs and 15 ECs will be explained in details. Table 9 reveals the satisfaction degree of CRs when the number of ECs is 10. It can be seen from Table 9 and Figure 6 that when the number of ECs is 10, the best regression order of stable solution sets is fourth-order. When the regression order is from the second-order regression to the fourth-order regression, $x_{1}$ , $x_{2}$ , $x_{3}$ , $x_{5}$ , $x_{6}$ , $x_{7}$ change. When the regression order is from the fourth-order regression to the tenth-order regression, the result remains unchanged. From the standpoint of standard deviations, the standard deviations between the solution sets of the adjacent regression model after Model 3 is 0 and remains stable. Table 10 reveals the satisfaction degree of CRs when the number of ECs is 15. It can be seen from Table 10 and Figure 7 that the best regression order of stable solution sets is ninth-order when the number of ECs is 15. When the regression order is from the second-order regression to the fourth-order regression and from fifth-order regression to ninth-order regression, $x_{1}$ , $x_{2}$ , $x_{3}$ , $x_{5}$ , $x_{7}$ change. When the regression order is from the fourth-order regression to fifth-order regression, $x_{1}$ , $x_{2}$ , $x_{3}$ , $x_{5}$ change. When the regression order is from the ninth-order regression to the fifteenth-order regression, the result remains unchanged. Simultaneously, the standard deviation gradually decreases with the increase of regression order. And when the regression order is from the eight-order regression to the ninth-order regression, the standard deviation is only 0.00011. The result remains unchanged after the ninth-order regression. The above simulation feasibility rate is above 95%, which proves its effectiveness.

The simulation results when the number of ECs is during 11–14 are not shown in detail here. This paper sorts out the best regression order when the number of ECs is from 9 to 15, as shown in Figure 8. It is obvious from Figure 8 that when the number of ECs increases, the regression order of the stable solution set also increases.

From the simulation results when the number of ECs is 15, it can be seen that when the regression order reaches the fifth-order, the standard deviation is only 0.0143. In order to simplify the model, the paper might as well regard the regression order when the standard deviation is less than 0.01 as the optimal regression order. From the standard deviation between Model 2 and 3 which solve the solution sets when the number of ECs is from 9 to 15 (the results is shown in Table 11), it is obverse that the standard deviation is less than 0.01 when the number of ECs is not more than 10, but the standard deviation is more than 0.01 when the number of ECs is more than 10. In order to ensure accuracy, the standard deviation between Model 3 and Model 4 needs to be considered when the number of ECs is from 11 to15. From the results of Table 12, it can be seen that the standard deviation is less than 0.01 when the number of ECs is from 11 to15. Therefore, this paper proposes a simple reference system based on the simulation results. When the number of ECs is less than or equal to 10, the optimal regression order can be the fourth-order. When the number of ECs is more than 10 and less than or equal to 15, the optimal regression order can be the fifth-order. It is clear from the results in Tables 11 and 12 that the reference can ensure that the standard deviation of the results is less than 0.01.

There are two reasons why the number of ECs considered in this study is limited to 15. One reason is that The LINGO software solution method is simplex algorithm. The basic idea of the simplex algorithm is: (1) find a vertex in the feasible region, and judge whether it is optimal according to certain rules; (2) if not, switch to another vertex adjacent to it, and make the objective function value better; (3) keep doing this until finding an optimal solution. When the number of ECs is excessive, it leads to insufficient system memory, and it may be impossible to arrive at an optimal solution. The other reason is that 15 ECs have met the number of ECs to be considered in the design process of most products. This is an experience from actual operation. If the quantity of ECs is too large, the numerical differentials among their outcomes will be minimal, and it will be difficult for users to choose the EC to focus on based on the gap. At the same time, the excessive number of ECs also increases the complexity of the product design process, which leads to a long design cycle. It is unfavorable from the perspective of company strategy.

6. Conclusions and future studies

In product design process, the feedback process is required to better capture the satisfaction degree of CRs achieved by the actual values of ECs. This process is needed to determine the priorities of ECs during improvement with limited resources. Wang et al. (2020) proposed the R-QFD method for examining the feedback process, explicitly addressing the feedback mechanism in the QFD. This method constitutes the foundational theoretical basis for R-QFD research. Building upon the research of Wang et al. (2020), this paper establishes an optimized model used in the R-QFD framework, achieving significant progress in the theoretical aspects of R-QFD. Firstly, this paper systematically reviews the ideas behind the establishment of the R-QFD model in Wang et al. (2020) and summarizes them into operational steps. Secondly, an upgrade to the R-QFD model is presented in this paper, incorporating the interrelationships among ECs. In order to solute the model, a simplified solution for optimization models with higher-order interrelationships was proposed and tested with simulation. Additionally, in terms of practical application, this paper introduces real cases from Japanese manufacturing companies to further substantiate the applicability of the R-QFD model.

The research outcomes of this paper, based on the feedback process of QFD, provide a novel theoretical tool for assessing the degree of goal accomplishment. In comparison to existing feedback tools, the model in this paper takes into account the interrelationships among EC, which improves the inference accuracy of R-QFD, and allows for the inclusion of a greater number of ECs in the consideration.

Several suggestions can be proposed based on the results.

The interrelationships among ECs greatly affect the accuracy of R-QFD models, so the higher-order interrelationships among ECs need to be considered in R-QFD. Only considering the relationships between ECs and CRs and the second-order interrelationships between every 2 ECs are not enough. This paper verifies the necessity of the optimization model to join the interrelationships among ECs through experiment design and give the solutions.
A simple instruction on the interrelationship examination was proposed based on simulation. When the number of ECs is less than 10, it is suggested that the interrelationships among ECs no more than four need to be considered in R-QFD. When the number of ECs is greater than 10 and less than or equal to 15, it is suggested that the interrelationships among ECs no more than five need to be examined in R-QFD. The proposal may facility companies to use R-QFD to feedback the implementation situation of the design plan for different numbers of ECs.

There are some limitations in this study which would be examined in the future research. The paper only tested the feasibility of the optimization models. The effectiveness of R-QFD with the optimization models needs to be further tested empirically. In addition, the instruction on the simple solutions is limited to 15 ECs. When proposing the simple solution for the higher-order interrelationships, the standard deviation changes were limited to no more than 1%. The future researches can extend the limitations to more ECs.

This work is supported by the National Natural Science Foundation of China (No.71572104).

Figure 1

A closed loop formulation model on QFD

[Figure omitted. See PDF]

Figure 2

The parameters in HOQ

[Figure omitted. See PDF]

Figure 3

Operation process flowchart of QFD and R-QFD

[Figure omitted. See PDF]

Figure 4

The satisfaction degree of CRs with 9 ECs

[Figure omitted. See PDF]

Figure 5

Standard deviations of 9 ECs used R-QFD model

[Figure omitted. See PDF]

Figure 6

Stability of 10 ECs solution sets

[Figure omitted. See PDF]

Figure 7

Stability of 15 ECs solution sets

[Figure omitted. See PDF]

Figure 8

The best regression order of 9–15 ECs

[Figure omitted. See PDF]

Table 1

Summary of literature researches

Areas	Research methods	Views	Reference
CRs capture and priorities	Fuzzy methods DCE	Reduce the accuracy of the CRs capture and priority	Khoo and Ho (1996), Temponi et al. (1999), Bottani and Rizzi (2006), Kwong and Bai (2002), Li et al. (2013)
ECs priorities	Fuzzy methods	Reduce the accuracy of ECs priority and ignore the interrelationships of ECs	Wang (1999), Karsak et al. (2003), Kahraman et al. (2006), Wu and Liao (2018)
Relationship matrix between CRs and ECs	Kano’s model Fuzzy methods	Kano’s model had not been well quantified. There is no unified and perfect plan for fuzzy rules	Tan and Shen (2000), Vanegas and Labib (2001), Chen and Weng (2006)
QFD optimization models	Fuzzy methods	The interrelationships among ECs were not quantified	Kim et al. (2000), Chen and Chen (2006), Tang et al. (2002)
Integrating QFD with other methods	Combine with TRIZ, AHP etc.	Only focus on the application of QFD in the forward process	Moskowitz and Kim (1997), Yamashina et al. (2002), Bhattacharya et al. (2005), Kubler et al. (2016)
R-QFD	Optimization models	Feedback process	Wang et al. (2020)